This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122913 #4 Jul 22 2025 16:20:50
%S A122913 1,1,3,1,6,2,3,1,6,5,3,4,12,2,6,1,3,10,15,5,9,5,18,25,9,13,9,14,12,7,
%T A122913 6,9,3,17,9,9,15,12,9,6,6,3,3,11,42,18,21,9,66,10,33,5,27,7,48,80,24,
%U A122913 40,12,20,6,10,3,5,3,7,3,79,75,63,96,40,48,20,24,10,12,5,6,15,3,22,72,11
%N A122913 Minimum numbers k such that (k^2*2^n + 1) is prime.
%C A122913 3 divides a(2k+1) for k>0. Corresponding primes of the form (k^2*2^n + 1) are listed in A122912[n] = {3,5,73,17,1153,257,1153,257,18433,25601,18433,65537,1179649,65537,1179649,65537,1179649,26214401,117964801,...}. There are repeating patterns in a(n) such that for many n a(n) = 2*a(n+2) and a(n+1) = 2*a(n+3). For example, {6,2,3,1}, {12,2,6,1}, {42,18,21,9}, {96,40,48,20,24,10,12,5,6}, {66,10,33,5}, {48,80,24,40,12,20,6,10,3}, {366,38,183,19}. These patterns correspond to identical twin runs in A122912[n] such that A122912[n] = A122912[n+2] and A122912[n+1] = A122912[n+3]. The final index of many such twin runs is perfect power such as {8,16,25,64,81,100,...}.
%F A122913 a(n) = Sqrt[ (A122912[n] - 1) / 2^n ].
%t A122913 mnk[n_]:=Module[{k=1,c=2^n},While[!PrimeQ[k^2 c+1],k++];k]; Array[mnk,90] (* _Harvey P. Dale_, Jul 22 2025 *)
%Y A122913 Cf. A112912.
%K A122913 nonn
%O A122913 1,3
%A A122913 _Alexander Adamchuk_, Sep 18 2006